This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard inner product. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring-Wirth non-expansive condition under non-standard metrics while preserving isometry. Building upon the novel vector transport operators, we extend the modified Polak-Ribi$\acute{e}$re-Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis.

翻译：本文提出两种创新的向量传输算子，基于Cayley变换，用于嵌入非标准内积的广义Stiefel流形。具体而言，引入微分收缩变换以及Cayley变换对微分矩阵指数的近似。这些向量传输被证明在非标准度量下满足Ring-Wirth非扩张条件，同时保持等距性。在新型向量传输算子的基础上，我们将修正的Polak-Ribière-Polyak (PRP) 共轭梯度法推广至广义Stiefel流形。在非单调线搜索条件下，我们证明该算法全局收敛至驻点。通过涉及广义特征值问题和典型相关分析的数值实验，经验性地验证了所提出的向量传输算子的有效性。